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Appendix A: Calculation of the non-linear electron number density

Inserting the nonlinear cooling rate Eq. (6) and the injection rate into Eq. (2) gives us the differential equation for the SSC cooled electron number density n_SSC (we drop the subscript in the following): (A.1)Multiplying the equation with γ²/A₀ we obtain with the definitions y = A₀t and S = γ²n(A.2)We yield with ξ = γ^-1(A.3)If we define the implicit time variable T through (A.4)the differential equation becomes (A.5)Formally multiplying this equation with dy   ! dT results in (A.6)This differential equation for the electron number density can be solved with the method of characteristics. Thus, we obtain (A.7)where ξ₀ = ξ − T is a constant of integration. Equation (A.7) can be easily integrated with respect to T, which results in (A.8)We now require that S(ξ = 0,T) = 0, which means (A.9)Collecting terms, we find S to be (A.10)Since our flare begins at T = 0, we are not interested in events that take place before that moment. Hence, we find the electron number density: (A.11)We see that we have two Heaviside functions defining upper limits for γ. It is an easy task to compare them, and to find out which one is lower than the other one. Having done so, we find the solution for the nonlinearly cooled electron number density to be (A.12)yielding Eq. (7).

Appendix B: Derivation of U

The time variable T has been defined through Eq. (8): (B.1) Inserting Eq. (A.10), we gain (B.2) As stated before, we are only interested in solutions for T ≥ 0. Hence, we can neglect the third Heaviside function, which results in (B.3)A first substitution w = ξ − T yields (B.4)while a second substitution w = Tv gives (B.5) where we re-substituted in the last step.

The purpose of the next two substitutions is to get rid of the time variable in the limits of the integral. In order to achieve this we first set x = γ₁T and introduce g₂ = γ₂/γ₁, which yields (B.6) Now, we use u = vx, resulting in (B.7)This integral can be expressed in terms of the hypergeometric function, but that would not yield an analytical form. Nonetheless, one can obtain an approximate solution in the regimes (small x), and x ≥ 1 (large x). An analytical continuation serves as a solution for the intermediate regime. For small x the integral can be written as (B.8) Similarly, we achieve for large x (B.9) The requirement for the solution of the intermediate regime is that it must be continuous, meaning , and U₂(1) = U₃(1). In order to accomplish such a behavior, we can first assume a proper solution U₂ with some unspecified constants, and then try to fit it to the boundary conditions. A good ansatz is (B.10)Matching the solution with the boundary conditions, yields the values of the constants a′, b′, and c′: Since we had only two equations for three parameters, we chose a′ = 1.

Thus, the obtained solution for the intermediate x-range is (B.11)Before we summarize the results, we need to say a few words about the spectral index s. We already stated that it must be greater than 1. But according to our results above, we also find that s ≠ 3. Thus, we have two different cases to consider: s > 3, and 1 < s < 3.

Collecting terms, we find for s > 3 (B.12)and for 1 < s < 3 (B.13)with , and .

Appendix C: The non-linear time variable x

Since , we can separate the variables obtaining . This can be integrated quite easily except for both intermediate cases. However, we can find approximative solutions by using the same approximations of these cases we used already during the calculation of the synchrotron spectra.

Beginning with the case s > 3 we yield (C.1)We require y(x = 0) = 0, which means c₁ = 0. The intermediate range becomes (C.2)

We approximated the denominator with the leading term. The validity of this approximation for most parts of the x-range can be seen in Fig. C.1. It becomes less valid for s → 3. For the third case we find (C.3)These equations can be inverted simply, yielding x(y). As for U(x), we require x to be continuous at the points and y₂ = y(x = 1). Matching the solutions at these points we find the values For the case 1 < s < 3 we find similarly (C.8)We require y(x = 0) = 0 again, which means c₄ = 0. The intermediate range becomes (C.9)

We approximated the denominator with the leading term. The validity of this approximation for most parts of the x-range can be seen in Fig.  C.2 . The largest errors occur for small values of  x.

The last case yields (C.10)As before, a simple inversion leads to x(y), while the requirement that the solution should be continuous at the points and y₄ = y(x = 1) gives the values

Appendix D: The implicit time variable of the combined cooling

The differential Eq. (2) may look a little bit more complicated with the combined cooling term (13). However, the solution can be obtained with the methods outlined in Appendix  A, yielding solution (14). The important difference is the definition of the implicit time variable, which has to be chosen as (D.1)Using the definition of U from Appendix B, this can be written as (D.2)Thus, we can use the previous results to obtain for s > 3 (D.3)and for 1 < s < 3 (D.4)where we defined , and used the leading term approximation discussed in Appendix C for the intermediate regimes.

D.1. Large spectral index

Similarly to the steps in Appendix C, we calculate the dependence . We begin with the case s > 3. For we find (D.5)As before, we set , since . Inverting Eq. (D.5) yields (D.6)Obviously, is found from the condition yielding (D.7)For we find (D.8)or the other way around (D.9)Since is supposed to be continuous, we find the constant by matching the solutions for resulting in (D.10)We also obtain from the condition (D.11)Defining we yield for (D.12)The problem arising is that we cannot find an inverted expression for . However, we can obtain approximative results for small and large arguments of the arctan-function.

In order to achieve these approximations we define the injection parameter (D.13)for which Eq. (D.12) becomes (D.14)For α₀ ≪ 1 the argument of the arctan-function is always (much) larger than unity, since . We, therefore, approximate , set , and obtain (D.15)This can be easily inverted, yielding the linear solution (D.16)Matching this solution with Eq. (D.9) yields (D.17)For α₀ ≫ 1 we have to consider two cases. If , we see that . Thus, we can approximate the arctan-function to third order as , resulting in (D.18)Inverting yields (D.19)with (D.20)and (D.21)In the case , we can approximate again arctan(x/α₀) ≈ π/2, yielding with (D.22)or inverted (D.23)with (D.24)What one can see here is that the injection parameter controls significantly the cooling behavior of the electrons. For α₀ ≪ 1 the solution is purely linear, while for α₀ ≫ 1 it is non-linear and becomes linear at later times, just as we expected.

Before we proceed with the case 1 < s < 3 we list the results of this section once more in a compact form. We also approximate the results for g₂ ≫ 1, which, as one will see, simplifies a lot. with and Since in this approximation , , and, thus, one can neglect .

D.2. Small spectral index

We will now derive the explicit form of the implicit time variable  for 1 < s < 3.

The first regime is , yielding (D.37)Since , obviously d₁ = 0, and the inversion becomes (D.38)with (D.39)The next time step is , resulting in (D.40)where we defined . For α₀ ≪ 1 we see that (as long as s is not too close to 1 or 3), and with we approximate the integral as (D.41)The inversion is easily performed, yielding (D.42)where we obtain by matching the solutions (D.43)and (D.44)For α₀ ≫ 1 we see that . As a rough approximation this is also much lower than , and, therefore, we achieve the integral (D.45)The inverted equation is (D.46)with (D.47)and (D.48)Similarly to the case for large spectral indices, we obtain here for the integral in that time regime (D.49)We will continue with the same approximations as before, yielding for α₀ ≪ 1 , and with the result (D.50)The inversion is obviously (D.51)where (D.52)For α₀ ≫ 1 we use for the approximation , yielding (D.53)Hence, (D.54)with (D.55)and (D.56)

The last case is for , where we can use the “linear” approximation arctan(x/α₀) ≈ π/2, again. With we achieve (D.57)or inverted (D.58)where we defined (D.59)As we did for the case of large spectral indices, we sum up our results in a short list, and perform the approximation for g₂ ≫ 1. with and

© ESO, 2010